On the scalar potential of minimal flavour violation

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Published for SISSA by Springer Received: April 22, 2011 Accepted: June 21, 2011 Published: July 4, 2011

On the scalar potential of minimal flavour violation

R. Alonso,^a M.B. Gavela,^a L. Merlo^b,c and S. Rigolin^d

aDepartamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid and Instituto de F´ısica Te´orica IFT-UAM/CSIC,

Cantoblanco, 28049 Madrid, Spain

bPhysik-Department, Technische Universit¨at M¨unchen, James-Franck-Strasse, D-85748 Garching, Germany

cTUM Institute for Advanced Study, Technische Universit¨at M¨unchen, Lichtenbergstrasse 2a, D-85748 Garching, Germany

dDipartimento di Fisica Galileo Galilei, Universit`a di Padova and INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padua, Italy

E-mail: [email protected],[email protected], [email protected],[email protected]

Abstract: Assuming the Minimal Flavour Violation hypothesis, we derive the general scalar potential for fields whose background values are the Yukawa couplings. We analyze the minimum of the potential and discuss the fine-tuning required to dynamically generate the mass hierarchies and the mixings between different quark generations. Two main cases are considered, corresponding to Yukawa interactions being effective operators of dimension five or six (or, equivalently, resulting from bi-fundamental and fundamental scalar fields, respectively). At the renormalizable and classical level, no mixing is naturally induced from dimension five Yukawa operators. On the contrary, from dimension six Yukawa operators one mixing angle and a strong mass hierarchy among the generations result.

Keywords: Beyond Standard Model, Quark Masses and SM Parameters, Global Symme- tries

ArXiv ePrint: 1103.2915

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Contents

1 Introduction 1

2 Two family case 5

2.1 d = 5 Yukawa operators: the bi-fundamental approach 6 2.1.1 The scalar potential at the renormalizable level 7 2.1.2 The scalar potential at the non-renormalizable level 10

2.2 d = 6 Yukawa operator: the fundamental approach 11

2.2.1 The scalar potential 13

2.2.2 The first generation 14

3 The three-family case 15

3.1 d = 5 Yukawa operator: the bi-fundamental approach 15

3.2 d = 6 Yukawa operator: the fundamental approach 17

3.3 Combining fundamentals and bi-fundamentals 18

4 Conclusions 19

A d = 6 operators in the bifundamental approach 21

B A fine-tuned scalar potential in the bifundamental approach 22

B.1 Minimization of the scalar potential 23

B.2 Three family case 25

C The scalar potential for the fundamental approach 28

D Note added in proof 29

1 Introduction

After years of intense searches, all flavour processes observed in the hadronic sector, from rare decays measurements in the kaon and pion sectors to superB-factories results, are well in agreement with the expectations of the Standard Model of particle physics (SM). To say that all flavour processes are consistent with the SM predictions is tantamount to state that all flavour effects observed until now are consistent with being generated through the Yukawa couplings, which are the sole vehicles of flavour and CP violation in the SM.

Nevertheless, the origin of fermion masses and mixings remains the most unsatisfactory question in the visible sector of nature: it involves important fine-tunings and lack of predictivity, as essentially for each mass or mixing angle a new parameter is added by

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hand to the SM. It is commonly expected that an underlying dynamics will provide a rationale for the observed patterns.

The hypothesis of Minimal Flavour Violation (MFV) [1] is a humble, matter-of-fact, and highly predictive working frame built only on: i) the assumption that, at low energies, the Yukawa couplings are the only sources of flavour and CP violation both in the SM and in whatever may be the flavour theory beyond it, abiding in this way to the experimental indications mentioned above; ii) the use of the flavour symmetries which the SM exhibits in the limit of vanishing Yukawa couplings.

Indeed, the hadronic part of the SM Lagrangian, in the absence of quark Yukawa terms, exhibits a flavour symmetry given by

Gf = SU(3)_QL× SU(3)_UR × SU(3)_DR, (1.1) plus three extra U(1) factors corresponding to the baryon number, the hypercharge and the Peccei-Quinn symmetry [2]. The non-abelian subgroup Gf controls the flavour structure of the Yukawa matrices, and we focus on it for the remainder of this paper. UnderGf, the SU(2)_L quark doublet,QL, and the SU(2)_L quark singlets,UR and DR, transform as:

QL∼ (3, 1, 1) , UR∼ (1, 3, 1) , DR∼ (1, 1, 3) . (1.2) The SM Yukawa interactions break explicitly the flavour symmetry:

LY =QLYDDRH + QLYUURH + h.c.˜ (1.3) The technical realization of the MFV ansatz promotes the Yukawa couplings YU,D to be spurion fields which transform under Gf as

YU ∼ (3, 3, 1) , YD ∼ (3, 1, 3) , (1.4) recovering the invariance under Gf of the full SM Lagrangian. Following the usual MFV convention for the Yukawas, one defines

YD =







yd 0 0 0 ys 0 0 0 yb





 , YU = V_CKM^†







yu 0 0 0 yc 0 0 0 yt





 , (1.5)

with V_CKM being the usual quark mixing matrix, encoding three angles and one CP-odd phase.

MFV is not a model of flavour and the value of the new dynamical flavour scale Λ_f is not fixed: it does not determine the energy scale at which new flavour effects will show up.

Nevertheless it is quite successful in predicting precise and constrained relations between different flavour transitions, to be observed whenever the new physics scale becomes ex- perimentally accessible [3]. The reason is that in the MFV framework the coefficients of all SM-gauge invariant operators have a fixed flavour structure in terms of Yukawa couplings, so as to make the operator invariant underGf, plus the fact that the top Yukawa coupling may dominate any coefficient in which it participates.¹

1This is modified, though, in some MFV versions such as two-Higgs doublet models [1] with extra discrete symmetries [4], or in models with strong dynamics [5].

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MFV sheds also an interesting light on the relative size of the electroweak and the flavour scale. The origin of all visible masses and the family structure are the two major unresolved puzzles of the SM and it is unknown whether a relation exists between the nature and size of those two scales. While the electroweak data, and the theoretical fine-tunings they require, suggest that new physics should appear around the TeV scale, traditional model-independent limits on the flavour scale Λ_f point to order(s) of magnitude larger values [6]. Within MFV both sizes could be reconciled instead around the TeV scale, due to the Yukawa suppression of the flavour-changing operator coefficients. This holds either assuming only the SM as the renormalizable theory [1] or in beyond the SM scenarios (BSM), such as supersymmetric [8] or extradimensional [9] versions of the MFV ansatz.²

It is unlikely that MFV holds at all scales [7]. MFV assumes a new dynamical scale Λ_f, which points to MFV being just an accidental low-energy property of the theory. In this sense, MFV implicitly points to a dynamical origin for the values of the Yukawa cou- plings. The latter may correspond to the vacuum expectation values (vevs) of elementary or composite fields or combinations of them. In other words, the spurions may be pro- moted to fields, usually called flavons. For instance, in the first formulation of MFV by Chivukula and Georgi [10], the Yukawa couplings corresponded to a fermion condensate.

In this work, we further explore the dynamical character of the flavons, in a rather model- independent way.

The Yukawa interactions may be then seen as effective operators of dimension larger than four — denominated Yukawa operators in what follows — weighted down by powers of the large flavour scale³ Λ_f. The precise dimension d of the Yukawa operators is not determined, as illustrated in figure1. As long as the vev to be taken by the flavon fields is smaller than Λf, an analysis ordered by inverse powers of this scale is a sensible approach.

The simplest case is that of ad = 5 operator:

LY =QL

Σd

Λ_fDRH + QL

Σu

Λ_fURH + h.c. ,˜ (1.6) with the scalar flavons Σ_d and Σu being dynamical fields in the bi-fundamental represen- tation of Gf (i.e. Σ_u ∼ (3, 3, 1) and Σd∼ (3, 1, 3), see eq. (1.4)) such that⁴

YD ≡ hΣ_di

Λ_f , YU ≡ hΣ_ui

Λ_f . (1.7)

2The BSM theory may introduce more than one distinct flavour scale: this work sticks to a conservative and minimalist approach, focusing on the physics related to Λf as described above.

3For instance, a possible realization among many takes Λf to be the mass of heavy flavour mediators in some BSM theory [11,12]: at energies E < Λf, they can be integrated out resulting in d > 4 operators involving the SM fields and the flavons.

4The Goldstone bosons that would result from the spontaneous breaking of a continuous global flavour symmetry, may be avoided for instance by gauging the symmetry. In practical realizations, this in turn tends to induce dangerous flavour-changing neutral currents mediated by the new gauge bosons. A new promising avenues to cope with this problem has been recently proposed in ref. [13,14].

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(Q

)

(D

)

H (Y

)

Figure 1. Effective Yukawa coupling.

An alternative realization, that we also explore below, is that of a d = 6 Yukawa operator, involving generically two scalar flavons for each spurion,

LY =QL

χ^L_dχ^R†_d

Λ²_f DRH + QL

χ^Luχ^R†u

Λ²_f URH + h.c. ,˜ (1.8) which provide the following relations between Yukawa couplings and vevs:

YD ≡ hχ^L_dihχ^R†_d i

Λ²_f , YU ≡ hχ^Luihχ^R†u i

Λ²_f . (1.9)

In this interesting case, the flavons are simply vectors under the flavour group, alike to quarks, with the simplest quantum number assignment beingχ^L_u,d∼ (3, 1, 1), χ^Ru ∼ (1, 3, 1) and χ^R_d ∼ (1, 1, 3). Following this pattern, would the Yukawa couplings result from a condensate of fermionic flavons [10], ad = 7 Yukawa operator could be adequate

YD ≡ hΨ^L_dΨ^R_di

Λ³_f , YU ≡ hΨ^L_uΨ^R_ui

Λ³_f , (1.10)

with fermions quantum numbers under Gf as in the previous case. Notice that these re- alizations in which the Yukawa couplings correspond to the vev of an aggregate of fields, rather than to a single field, are not the simplest realization of MFV as defined in ref. [28], while still corresponding to the essential idea that the Yukawa spurions may have a dy- namical origin.

The goal of this work is to address the problem of the determination of the general scalar potential, compatible with the flavour symmetry Gf, for the flavon fields denoted above by Σ orχ. An interesting question is whether it is possible to obtain the SM Yukawa pattern — i.e. the observed values of quark masses and mixings — with a renormalizable potential. We derive the potential, analyze the possible vacua, and discuss the degree of

“naturalness” of the possible solutions. It will be shown that the possibility of obtaining a large mass hierarchy and mixing at the renormalizable level varies much depending on the dimension of the Yukawa operator. The role played by non-renormalizable terms and the fine-tunings required to accommodate the full spectrum will be explored.

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A relevant issue is what will be meant by natural: following ’t Hooft’s naturalness criteria, all dimensionless free parameters of the potential not constrained by the symmetry should be of order one, and all dimensionful ones are expected to be of the order of the scale(s) of the theory. We will thus explore in which cases — if any — those criteria allow that the minimum of the MFV potential corresponds automatically to mixings and large mass hierarchies. Stronger than O(10%) adjustments (typical Clebsh-Gordan values in any theory) will be considered fine-tuned.

It is worth to note that the structure of the scalar potentials constructed here is more general than the particular effective realization in eqs. (1.6) and (1.8). Indeed, it relies exclusively on invariance under the symmetry Gf and on the flavon representation, bi- fundamental or fundamental.⁵

We limit our detailed discussions below to the quark sector. The implementation of MFV in the leptonic sector [15,16] requires some supplementary assumptions, as Majorana neutrino masses require to extend the SM and involve a new scale: that of lepton number violation. Due to the smallness of neutrino masses, the effective scale of lepton number violation must be distinct from the flavour and electroweak ones, if new observable flavour effects are to be expected [17]. Nevertheless, the analysis of the flavon scalar potential performed below may also apply when considering leptons, although the precise analysis and implications for the leptonic spectrum will be carried out elsewhere.

The structure of the manuscript is as follows. In section 2, for the two-family case we analyze the renormalizable potential for d = 5 and d = 6 Yukawa operators, or in other words of flavons in the bi-fundamental and in the fundamental of Gf, respectively, showing that in the latter case mixing and a strong hierarchy are intrinsically present.

The corrections induced by non-renormalizable terms are also discussed. In section 3 the analogous analyses are carried out for the realistic three-family case and it is also discussed the qualitative new features appearing when considering simultaneously d = 5 and d = 6 Yukawa operators. The conclusions are presented in section 4. Details of the analytical and numerical discussions of the potential minimization can be found in the appendices.

2 Two family case

We start the discussion of the general scalar potential for the MFV framework by illus- trating the two-family case, postponing the discussion of three families to the next section.

Even if we restrict to a simplified case, with a smaller number of Yukawa couplings and mixing angles, it is a very reasonable starting-up scenario, that corresponds to the limit in which the third family is decoupled, as suggested by the hierarchy between quark masses and the smallness of the CKM mixing angles⁶θ23andθ13. In this section, moreover, we will introduce most of the conventions and ideas to be used later on for the three-family analysis.

5For instance, the potential and the consequences for mixing obtained in this work will apply as well to the construction in ref. [13], notwithstanding the fact that there the flavon vevs show and inverse hierarchy than that for the minimal version of MFV, as they are proportional to the inverse of the SM Yukawa couplings.

6We follow in this paper the PDG [18] conventions for the CKM matrix parametrization.

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With only two generations the non-Abelian flavour symmetry group,Gf, is reduced to Gf = SU(2)_QL× SU(2)_UR × SU(2)_DR, (2.1) under which the quark fields transform as

QL∼ (2, 1, 1) , UR∼ (1, 2, 1) , DR∼ (1, 1, 2) . (2.2) Following the MFV prescription, in order to preserve the flavour symmetry in the La- grangian, the Yukawa spurions introduced in eq. (1.4) now transform under Gf as

YU ∼ (2, 2, 1) , YD ∼ (2, 1, 2) . (2.3) The masses of the first two generations and the mixing angle among them arise once the spurions take the following values:

YD = yd 0 0 ys

!

, YU = V_C^† yu 0 0 yc

!

, (2.4)

where

V_C = cosθ sin θ

− sinθ cos θ

!

(2.5)

is the usual Cabibbo rotation among the first two families.

2.1 d = 5 Yukawa operators: the bi-fundamental approach

The most intuitive approach, in looking for a dynamical origin of MFV, is probably to promote each Yukawa coupling from a simple spurion to a flavon field. In other words, to consider the effective d = 5 Lagrangian described above in eq. (1.6). The new fields — flavons — are singlets under the SM gauge group but have, for the two-family case, the non-trivial transformation properties underGf given by

Σ_u ∼ (2, 2, 1) −→ Σ⁰u = Ω_LΣ_uΩ^†_U

R , Σ_d∼ (2, 1, 2) −→ Σ⁰d= Ω_LΣ_dΩ^†_D

R, (2.6) where Ω_X denotes the doublet transformation under the SU(2)_X-component of the flavour group. Once these flavon fields develop vevs as in eq. (1.7) and eq. (2.4), the flavour symmetry is explicitly broken and quark masses and mixings are originated. The effective field theory obtained at the electroweak scale is exactly MFV [1] (restricted to the two- family case). Then, within this approach, the problem of the origin of flavour is replaced by the need to explain if and how this particular vev configuration can naturally arise from the minimization of the associated scalar potential.

This minimal framework can be easily extended in different ways, such as, for instance:

• Considering different scales for the Σ_u and Σ_dflavon vevs.

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• Adding new representations. The most straightforward way to complete the basis in eq. (2.6), is to add a third flavon transforming as a bi-fundamental of the RH components:

ΣR∼ (1, 2, 2) −→ Σ⁰R= ΩURΣRΩ^†_D

R. (2.7)

This new field does not contribute to the Yukawa terms, at least at the renormaliz- able level, but introduces new operators with respect to MFV, which induce flavour changing neutral currents (FCNC) mediating fully right-handed (RH) processes.⁷

• Adding new replicas of the bi-fundamental representations. This could be very helpful as a natural source of new scales and possible mixings.

The first two possibilities do not affect essentially the flavour structure of the quark Yukawa couplings, which is the focus of this work, and we will not consider them below. No further consideration is given either in this section to the third possibility, both for the sake of simplicity and because of the aesthetically unappealing aspect of being a trivial replacement of the puzzle of quark replication with that of flavon replication.

We will thus restrict the remaining of this section to the analysis of the potential for just one Σu and one Σdfields, eqs. (2.6). The general scalar potential, can then be written as a sum of two parts, the first dealing only with the SM Higgs fields and the second accounting also for the flavons interactions:

V ≡ VH +VΣ= −µ²H^†H + λH(H^†H)²+

∞

X

i=4

V⁽ⁱ⁾[H, Σu, Σd]. (2.8)

Inside V⁽ⁱ⁾ all possible scalar potential terms of the effective field theory are included. In particular, V⁽⁴⁾ contains all the renormalizable couplings written in terms of H and Σu,d

while V^(i>4) incorporate all the non-renormalizable higher dimensional operators. There is no particular reason to impose that the EW and the flavour symmetry breaking should occur at the same scale. Indeed it is plausible that the flavour symmetry is broken by some new physics mechanism at a larger energy scale. Although it is true that the mixed Higgs- flavons terms could affect the value and location of the electroweak and flavour minima, the flavour composition of each term will not be modified by them. Once the flavour symmetry breaking occurs, all the terms inV⁽ⁱ⁾ either contribute to the scalar potential as constants or can be redefined into µ² or λH. In what follows the analysis is restricted to consider only the flavon part of the scalar potential,V⁽ⁱ⁾[Σu, Σd].

2.1.1 The scalar potential at the renormalizable level

From the transformation properties in eq. (2.6), it is straightforward to write the most gen- eral independent invariants that enter in the scalar potential. At the renormalizable level

7The phenomenological impact of these operators has already been introduced and studied in the three- family case in ref. [19,20], in a different context.

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and for the case of two generations, five independent invariants can be constructed⁸ [21]:

Au = Tr

Σ_uΣ^†_u , Bu = det (Σ_u), Ad= Tr

Σ_dΣ^†_d , Bd= det (Σ_d) , Aud= Tr

Σ_uΣ^†_uΣ_dΣ^†_d .

(2.9)

eqs. (1.7) and (2.4) allow to express these invariants in terms of physical observables, i.e.

the four Yukawa eigenvalues and the Cabibbo angle:

hΣ_di = Λ_f yd 0 0 ys

!

, hΣ_ui = Λ_fV_C^† yu 0 0 yc

!

, (2.10)

leading to:

hAui = Λ²_f(y_u²+y_c²), hBui = Λ²_fyuyc, hAdi = Λ²_f(y²_d+ys²), hBdi = Λ²_fydys, hAudi = Λ⁴_f

y²c−y²u

ys²−y_d²
cos 2θ + yc²+yu²

ys²+y²_d


/2 .

(2.11)

Notice that the mixing angle appears only in the vev ofAud, which is the only operator that mixes the up and down flavon sectors. This is as intuitively expected: the mixing angle describes the relative misalignment between two different directions in flavour space. It is also interesting to notice that the expression for hAudi is related to the Jarlskog invariant for two families,

4J = 4 dethYUY_U^†, YDY_D^†i

= (sin 2θ)² y²c−yu²

2

ys²−yd²

2

, by the following relation:

1 Λ⁴_f

∂hAudi

∂θ = −2√

J . (2.12)

Using the invariants in eqs. (2.9), the most general renormalizable scalar potential allowed by the flavour symmetry reads:

V⁽⁴⁾ =X

i=u,d



−µ²_iAi− ˜µ²_iBi+λiA²_i+ ˜λiB²_i

+gudAuAd+fudBuBd+X

i,j=u,d

hijAiBj+λudAud, (2.13) where strict naturalness criteria would require all dimensionless couplings λ, f, g, h to be of order 1, and the dimensionful µ-terms to be smaller or equal than Λf although of the same order of magnitude. It is clear from the start that, with the only use of symmetry implemented here, a strict implementation of such criteria could lead at best to a strong hierarchy with some fields massless and the rest with masses of about the same scale. The “fan” structure of quark mass splittings observed clearly calls, instead, for a readjustment of the relative size of some µ parameters, at least when restraining to the

8Any other invariant operator can be expressed in terms of these five independent invariants. For example: T r`ΣuΣ<sup>†</sup><sub>u</sub>ΣuΣ<sup>†</sup><sub>u</sub>´ = T r `ΣuΣ^†_u´2

− 2 det (Σu)².

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analysis of the renormalizable and classic terms of the potential. One question is whether, in this situation, even further fine-tunings are required among the mass parameters in the potential to accommodate nature.

The relations in eq. (2.11) allow to determine the positions of the potential minima in terms of physical observables. A careful analytical and numerical study of the potential can be found in the appendices. Here we briefly comment on the most relevant physical results. Consider first the angular part of the potential. Deriving V⁽⁴⁾ with respect to the angle θ, it follows that

∂V⁽⁴⁾

∂θ     min

≡λud∂hAudi

∂θ ∝λudsin 2θ y²c−yu²

ys²−y²d
∝λud

√J . (2.14)

The minimum of the scalar potential thus occurs when at least one of the following con- ditions is satisfied i) λud = 0, ii) sinθ = 0, iii) cos θ = 0 or iv) two Yukawas in the same sector are degenerate. When condition i) is imposed, the angle remains undetermined; this assumption corresponds however to a severe fine-tuning on the model, as no symmetry protects this term from reappearing at the quantum level. Instead, due to the smallness of the Cabibbo angle, condition ii) can be interpreted as a first order solution which needs to be subsequently corrected, for example by the introduction of higher order operators.

This possibility will be discussed in more detail in the next subsection. Finally, the last conditions, iii) and iv), are phenomenologically non representative of nature and large (higher order) corrections should be advocated in order to diminish the angle or to split the Yukawa degeneracy, respectively, making these solutions unattractive. All in all, the straightforward lesson that follows from eq. (2.14) is that, given the mass splittings ob- served in nature, the scalar potential for bi-fundamental flavons does not allow mixing at leading order.

From the requirement that the derivatives of the scalar potential with respect toyu,d,c,s

also vanish at the minima, four additional independent relations on the physical parameters are obtained. As discussed above, to obtain simultaneously a sizeable mixing and a mass spectrum largely splitted in masses, instead of generically degenerate, it is necessary to (re-)introduce a large, and unnatural, hierarchy among the different operators appearing in the scalar potential (see appendix B for numerical details).

These observations can be summarized stating that, with a natural choice of the co- efficients appearing in the renormalizable scalar potential V⁽⁴⁾, after minimization one naturally ends up with a vanishing or undetermined mixing angle and with a naturally de- generate spectrum. In this respect we agree with a remark that can be found in refs. [13, 21]. It is, however, interesting to notice that if the invariants Bu,d (i.e. the de- terminants of the flavons) are neglected, which could be justified for example introducing some ad hoc discrete symmetry, the minima equations would then allow, instead, solutions non-degenerate in mass for same-charge quarks, with (non-)vanishing Yukawa couplings for the first (second) quark generations. This may open the possibility to study a modified version of the scalar potential in eq. (2.13), that predicts a natural hierarchy among the Yukawas of different generations.

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2.1.2 The scalar potential at the non-renormalizable level

Consider the addition of non-renormalizable operators to the scalar potential, V^(i>4). It is very interesting to notice that this does not require the introduction of new invariants beyond those in eq. (2.9): all higher order traces and determinants can in fact be expressed in terms of that basis of five “renormalizable” invariants.

The lowest higher dimensional contributions to the scalar potential have dimension six (the complete list can be found in appendix A). At this order, the only terms affecting the mixing angle are

V⁽⁶⁾⊃ 1 Λ²_f

X

i=u,d

(αiAudBi+βiAudAi) . (2.15)

These terms, however, show the same dependence on the Cabibbo angle previously found in eq. (2.14) and, consequently, they can simply be absorbed in the redefinition of the lowest order parameter, λud. In other words, even at the non-renormalizable level, the most favorable trend leads to no mixing. To find a non-trivial angular structure it turns out that terms in the potential of dimension eight (or higher) have to be considered, that is

V⁽⁸⁾⊃λududA²_ud, (2.16)

and eq. (2.14) would be replaced by

∂V

∂θ    min

∝ sin 2θ y_c²−y_u²

y²_s−y²_d


λud− 2y_c²y_s²λududsin²θ + . . .  , (2.17)

implying

sin²θ ' λud

2y²cy²sλudud

. (2.18)

Using the experimental values of the Yukawa couplings ys and yc, a meaningful value for sinθ can be obtained although at the price of assuming a highly fine-tuned hierarchy between the dimensionless coefficients ofd = 4 and d = 8 terms, λud/λudud ∼ 10⁻¹⁰, that cannot be naturally justified in an effective Lagrangian approach.

The remaining four equations defining the minima, obtained deriving the scalar po- tential with respect to yu,d,c,s, lead to no improvement as compared to the renormalizable case: the Yukawa couplings are always given by general combinations of the coefficients of the scalar potential, underlining the complete absence of hierarchies among them. Realistic masses can be obtained at the classical level only when suitable fine-tunings are enforced.⁹ To summarize, it is possible to account for a non-vanishing mixing angle adding non- renormalizable terms to the scalar potential, although at the prize of introducing a large fine-tuning. This requirement comes in addition to the fact that the hierarchies among the Yukawa couplings can only be imposed by hand. Therefore the use of bi-fundamental scalar fields leads to an unsatisfactory answer to the problem of explaining the origin of flavour within the MFV hypothesis.

9See note added in proof.

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For the sake of illustrating the argument with a practical exercise, we conclude this section showing, as an explicit example, a fine-tuned scalar potential which can allow hierarchical Yukawas and a non-vanishing mixing angle:

V =X

i

−µ²iAi+ ˜λiBi²+λiA²i

 +λudud

Λ⁴_f (Audud−2Auudd)−bµ˜²dBd−uµ˜²uBu+θλudAud, (2.19) where u,d,θ are suppressing factors, possibly associated to some discrete symmetry, and Auudd,Audud, dimension eight invariants defined by the following relations:

Audud= Tr

Σ_uΣ^†_uΣ_dΣ^†_dΣ_uΣ^†_uΣ_dΣ^†_d , Auudd= Tr

Σ_uΣ^†_uΣ_uΣ^†_uΣ_dΣ^†_dΣ_dΣ^†_d . (2.20) By minimizing the potential in eq. (2.19) one obtains the following values for the Yukawa eigenvalues and the Cabibbo angle:

yu'u

√λuµ˜u

√2 ˜λuµu

µ˜u

Λ_f , yd'd

√λdµ˜d

√2 ˜λdµd

µ˜d

Λ_f , yc' µu

√2 Λ_f√ λu

, ys' µd

√2 Λ_f√ λd

,

sin²θ ' θ λud

λududy²cys²

.

(2.21)

Imposing for no good reason the values u ∼ 10⁻³, d ∼ 5 × 10⁻², θ ∼ 10⁻¹⁰ and µ/(√

λΛf) ≈ ˜µ/(p˜λΛf) ∼ 10⁻³, the correct hierarchies between the quark masses and the correct Cabibbo angle could be obtained (see details for this special case in appendix B).

The discussion about d = 8 terms presented above has pure illustrative purposes, as it may be a priori misleading to discuss the effects ofd = 8 terms in the potential without simultaneously considering quantum or other higher-order sources of corrections, such as the possible impact of a Σ_R flavon¹⁰ — see eq. (2.7) — or otherGf representations.

2.2 d = 6 Yukawa operator: the fundamental approach

The identification of the Yukawa spurions as single flavon fields, transforming in the bi- fundamental representation of the flavour group (e.g. for ad = 5 Yukawa operator), is only one of the possible ways the MFV ansatz can be implemented. An attractive alternative is to consider the Yukawas as composite objects or aggregates of several fields, e.g. suggesting Yukawa operators withd > 5. In the simplest case, each Yukawa corresponds to two scalar fields χ transforming in the fundamental representation of Gf (e.g. Y ∼ hχihχ^0†i/Λ²_f, see eqs. (1.8) and (1.9)). This approach would a priori allow to introduce one new field for each component of the flavour symmetry: i.e. to reconstruct the spurions in eq. (1.4) just out of three vectors transforming as (2, 1, 1), (1, 2, 1) and (1, 1, 2). However, such a minimal setup leads to an unsatisfactory realization of the flavour sector as no physical mixing angle is

10The impact of the fully RH bi-fundamental ΣRis negligible: indeed it can enter in the scalar potential only as powers of ΣRΣ^†_Ror its hermitian conjugate, and in particular, being a singlet of SU(2)Q_L, it cannot mix with the other flavons. As a result, its contributions can always be absorbed through a redefinition of the parameters and then the conclusions above still hold.

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allowed at the renormalizable level.¹¹ The situation improves qualitatively, though, if two (2, 1, 1) representations are introduced, one for the up and one for the down quark sectors.

Consider then the following four fields:

χ^Lu ∈ (2, 1, 1) , χ^Ru ∈ (1, 2, 1) , χ^Ld ∈ (2, 1, 1) , χ^Rd ∈ (1, 1, 2) . (2.22) The corresponding d = 6 effective Lagrangian and Yukawa couplings have been shown in eqs. (1.8) and (1.9). These flavons are vectors under the flavour symmetry. The only physical invariants that can be associated to vectors are the norm of the vectors and, eventually, their relative angles. Any matrix resulting from multiplying two vectors has only one non-vanishing eigenvalue, independently of the number of dimensions of the space.

This fact alone already implies that, at the leading renormalizable order under discussion, just one “up”-type quark and one “down”-type quark are massive: a strong mass hierarchy between quarks of the same electric charge is thus automatic in this setup, which is a very promising first step in the path to explain the observed quark mass hierarchies.

More in detail, the resulting Yukawa matrices are general 2 × 2 matrices, containing many unphysical parameters. Without loss of generality, it is possible to express the Yukawa couplings in terms of physical quantities by choosing the flavon vevs as follows:

hχii ≡ |χi| V_i 0 1

!

, (2.23)

where by |χi| we denote the norm of the vev of χ, |χi| ≡ |hχii|, and V_i are 2 × 2 unitary matrices. Redefining the quark fields as follows,

Q⁰_L= V_L^(d)†QL, U_R⁰ = V_R^(u)†UR, D_R⁰ = V_R^(d)†DR, (2.24) it results

LY =Q⁰LYDD⁰RH + Q⁰LYUUR⁰H + h.c. ,˜ (2.25) with the corresponding Yukawa matrices given by¹²

YD =  χ^L_d

  χ^R_d

 Λ²_f

0 0 0 1

!

, YU =

 χ^Lu

   χ^Ru

Λ²_f V_L^(d)†V_L^(u) 0 0 0 1

!

. (2.26)

This illustrates explicitly that: i) there is a natural hierarchy among the mass of the first and second generations, without imposing any constraint on the parameters of the scalar potential ; ii) the product V_L^(d)†V_L^(u) is a non-trivial unitary matrix that contains all the information about the mixing angle (the phase can be easily removed in the two-family case under discussion). There is now a clear geometrical interpretation of the Cabibbo angle: the mixing angle between two generations of quarks is the misalignment of the χ^L

11 Because then the flavons associated to the up and down left-handed character are not misaligned in flavour space, but correspond instead to just one (2, 1, 1) flavon.

12The cutoff scale Λf refers to the scale of the flavour dynamics. In principle we could have different scales for the left and right flavons as well as for the up and down ones, but for simplicity we assume that all the scales are close and Λf refers to the average value.

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flavons in the flavour space, with the mixing matrix appearing in weak currents, eq. (2.5), given by

V_C = V_L^(u)†V_L^(d). (2.27)

Let us compare the phenomenology expected from bi-fundamental flavons (i.e. d = 5 Yukawa operator) with that from fundamental flavons (i.e. d = 6 Yukawa operators).

For bi-fundamentals, the list of effective FCNC operators is exactly the same that in the original MFV proposal [1]. The case of fundamentals presents some differences: higher- dimension invariants can be constructed in this case, exhibiting lower dimension than in the bi-fundamental case. For instance, one can compare these two operators:

DRΣ^†_dΣuΣ^†_uQL∼ [mass]⁶ ←→ DRχ^Rd χ^L†u QL∼ [mass]⁵, (2.28) where the mass dimension of the invariant is shown in brackets; with these two types of basic bilinear FCNC structures it is possible to build effective operators describing FCNC processes, but differing on the degree of suppression that they exhibit. This underlines the fact that the identification of Yukawa couplings with aggregates of two or more flavons is a setup which goes technically beyond the realization of MFV, resulting possibly in a distinct phenomenology which could provide a way to distinguish between fundamental and bi-fundamental origin

2.2.1 The scalar potential

The general scalar potential that can be written including flavons in the fundamental is analogous to that in eq. (2.8), replacing Σi withχi,

V ≡ VH +Vχ. (2.29)

Previous considerations regarding the scale separation between EW and flavour breaking scale hold also in this case, and in consequence the Higgs sector contributions will not be explicitly described.

Any flavour invariant operator can be constructed out of the following five independent building blocks:

χ^L†u χ^Lu, χ^R†u χ^Ru , χ^L†_d χ^Ld, χ^R†_d χ^Rd , χ^L†u χ^Ld. (2.30) From the expressions for the Yukawa matrices in eqs. (2.26), it follows that in this scenario the scalar potential depends only on three of the five physical parameters: one angle and the two (larger) Yukawa couplings

 χ^L_u

  χ^R_u

= Λ²_fyc,  χ^L_d

  χ^R_d

= Λ²_fys, χ^L†_u χ^L_d = cosθc

 χ^L_u

  χ^L_d

 , (2.31) given by the product of the left and right up (down) flavon moduli. As expected, the mixing angle is simply the angle defined in flavour space by the up and down left vectors.

From the point of view of the measurable quantities, there is a certain parametrization freedom, and a possible convenient choice is given by¹³

 χ^Ru

Λ_f = 1 =  χ^R_d

Λ_f . (2.32)

13See appendix C for a detailed discussion.

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As a result, the invariants physically relevant for the flavour structure are:

 χ^Lu

= Λ_fyc,  χ^L_d

= Λ_fys, χ^L†u χ^L_d = Λ²_fycys cosθ . (2.33) At the renormalizable level, the scalar potential is given by

V⁽⁴⁾ = − X

i=u,d

µ²_i χ^L†_i χ^L_i − X

i=u,d

µ˜²_i χ^R†_i χ^R_i −µ²_udχ^L†_u χ^L_d +. . . , (2.34)

where dots stand for all possible quartic couplings. The total number of operators that can be introduced at the renormalizable level is 20. However, as shown in appendix C, many of them (i.e. quartic couplings that mix different flavours) do not have any real impact on the existence and determination of the minima. Studying the latter, the following relations between the (large) up and down Yukawa eigenvalues and the Cabibbo angle follow:

ys²

yc²

= µ²_dλu

µ²uλd, cosθ =

√λuλdµ²_ud λudµuµd

(2.35) which shows that without strong fine-tunings this scenario can explain the hierarchy be- tween the first and second family, and account for a sizable Cabibbo angle.

2.2.2 The first generation

In this two-generation analysis, the first family has remained massless at the renormaliz- able level. A first possibility is that non-renormalizable corrections may induce this small masses. Non-renormalizable interactions manifest themselves in form of higher order con- tributions to the Yukawa operators and the flavon vevs and/or as non-renormalizable terms in the potential, which can modify its minima.

From eq. (1.8) and the flavon transformation properties, it follows that higher order contributions to the Yukawa operators can only be constructed by further insertions ofχ^†χ inside the renormalizable operators. However, such kind of insertions do not modify the flavour structure of the Yukawa matrices, but simply redefine the two heavier couplings,yc

andys. On the other hand, the introduction of higher order operators in the scalar potential has the effect of modifying the vevs of the flavons, replacing the relation in eq. (2.23) with

hχ^L,R_u,d i Λf

≡  

(1 + O())χ^L,R_u,d   



V_L,R^(u,d)(1 + O()) O() 1

!

, (2.36)

where   1 parametrizes the ratio among higher and leading order contributions. The only effect of these modifications is to redefine the mixing angle θ and the second family Yukawas, yc and ys, without changing the rank of the Yukawa matrices and leaving thus the first generation massless. In summary, non-renormalizable interactions cannot switch on additional (first family) Yukawas if they were absent at the renormalizable level.

An alternative can be built on the fact that each up-down set of fundamental flavons provides a supplementary scale, in addition to new sources of mixing from their mis- alignment. A possibility along this direction is to enlarge the number of flavons to six, made out of a set of three (χ^R_u,d plus just one χ^L) replicated: in total two χ^L ∼ (2, 1, 1),

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two χ^Ru ∼ (1, 2, 1) and two χ^R_d ∼ (1, 1, 2). In this case the Yukawa terms change in a non-trivial way:

YD ≡ P

ijα^d_ijhχ^L_iihχ^R†_j i

Λ²_f , YU ≡

P

ijα^u_ijhχ^L_iihχ^R†_j i

Λ²_f , (2.37)

with αi,j numerical coefficients and i, j running over all flavons. An explicit computation reveals that, for generic values of αij(6= 0), the rank of the Yukawa matrices is indeed two.

However, in this case, the natural hierarchy between the first and second family is lost, being all the Yukawas of the same order unless the vevs of the new flavons are unnaturally smaller than those of the first replica. In conclusion, adding new RH flavon copies does not lead either to an appealing and natural source of masses for the first generation.

3 The three-family case

Let us extend the previous analysis to the three-family case. While most of the procedure, with both bi-fundamental and fundamental representations, follows straightforwardly, two main differences should be underlined. First of all, the top Yukawa coupling, yt, is now a parameter which is of O(1). The fact that in the two-family case the largest Yukawa, yc was much smaller than one, allowed us to safely retain only the lowest order terms in the (Yukawa) perturbative expansion. In the three-family scenario, in principle, one should include all orders in the expansion. However, in this case, the Cayley-Hamilton identity [22,23] provides a way out, as it proves that a general 3 × 3 matrixX must satisfy the relation:

X³− Tr[X] X²+1

2X Tr[X]²− Tr

X²
− det[X] = 0 , (3.1) which allows to express all powers Xⁿ (with n > 2) in terms solely of 1, X and X², with coefficients involving the traces of X and X² and the determinant ofX. In the case under study, X corresponds to the invariant products Σ^†Σ or χ^†χ, depending on whether bi-fundamental or fundamental representations are considered.

The second main difference with respect to the two-family case, is the appearance of a physical phase in the quark mixing matrix. For the sake of simplicity, in this paper we disregard CP-violation, deferring its discussion to a future work [24].

3.1 d = 5 Yukawa operator: the bi-fundamental approach

In this section we extend the approach discussed in section 2.1 to the three-family case.

Consider two bi-triplets under the flavour symmetryGf, see eq. (1.4), Σu ∼ (3, 3, 1) −→ Σ⁰u = Ω_LΣuΩ^†_U

R , Σ_d∼ (3, 1, 3) −→ Σ⁰d= Ω_LΣ_dΩ^†_D

R, (3.2) where now the Ω_X matrices refer to the triplet transformations under the SU(3)_X compo- nent of the flavour group. The Yukawa Lagrangian is the same as that in eq. (1.6). Once the flavons develop a vev as in eq. (1.7), the flavour symmetry is broken and one recovers the observed fermion masses and CKM matrix given in eq. (1.5). Recall that the present realiza- tion is the simplest realization of the original MFV approach [1]. Again, it would be possible

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to extend it introducing a third RH flavon field, Σ_R∼ (1, 3, 3) −→ Σ⁰_R= Ω_URΣ_RΩ^†_D

R. We do not further consider it when constructing the scalar potential, as it cannot contribute to the Yukawa spurions neither at O(1/Λf) nor O(1/Λ²_f), that is, neither via d = 5 nor d = 6 Yukawa operators.

Restricting the explicit analysis to the part of the renormalizable scalar potential not containing the SM Higgs field, a complete and independent basis is given by the following seven invariant operators:

Au = Tr

Σ_uΣ^†_u , hAui = Λ²_f yt²+y²c+yu²

, Bu = det (Σ_u) , hBui = Λ³_fyuycyt,

Ad= Tr



Σ_dΣ^†_d , hAdi = Λ²_f yb²+y²s+yd²

, Bd= det (Σ_d) , hBdi = Λ³_fydysyb,

Auu= Tr



ΣuΣ^†_uΣuΣ^†_u , hAuui = Λ⁴_f y_t⁴+y⁴_c+y_u⁴
, Add= Tr

Σ_dΣ^†_dΣ_dΣ^†_d , hAddi = Λ⁴_f yb⁴+y⁴s+yd⁴

, Aud= Tr

Σ_uΣ^†_uΣ_dΣ^†_d , hAudi = Λ⁴_f(P0+Pint) ,

(3.3)

whereP0 and Pint encode the angular dependence, P0 ≡ −X

i<j

yu²i−yu²j y²di−y²_dj

sin²θij, (3.4)

Pint ≡ X

i<j,k

y²di−y²dk

yu²j −yu²k



sin²θiksin²θjk +

− y²d−ys²

yc²−y²t
sin²θ12sin²θ13sin²θ23 + (3.5)

+1

2 y²_d−y_s²

y²_c−y_t²
cosδ sin 2θ12sin 2θ23sinθ13,

with i, j, k = 1, 2, 3. P0 generalizes the expression found in the two-family case — see eq. (2.11) — containing all the terms with a single angular dependence. The second piece, instead, Pint, contains all contributions that involve more than one mixing angle. Notice that in this case the Jarlskog invariant appears only at the non-renormalizable level.

The most general scalar potential at the renormalizable level is now given by V⁽⁴⁾= X

i=u,d

−µ²iAi+ ˜µiBi+λiA²i +λ⁰iAii
+gudAuAd+λudAud. (3.6)

Notice that the invariants Bu,d have mass dimension three (instead of two for the two- generation case), so that no B_u,d² term can be introduced at this level.

The solutions that minimize this scalar potential have a pattern very similar to that in the two-family case: i) no mixing is favored,¹⁴ ii) in most of the parameter space. Now

14However, due to the peculiar structure of the last term in eq. (3.6), minima with non-vanishing angles are now allowed, although leading to solutions which are both fine-tuned and overall physically incorrect.

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however, there is a region in parameter space for which a hierachical solution is allowed for non strictly zero, but constrained, ˜µ. This solution has one non-vanishng Yukawa eigenvalue per up and down sectors, but to recover the hierarchy among top and bottom masses it is necessary to further demand gud < y_b²/y_t² which, in the absence of ad hoc symmetries, results in a similar degree of fine-tunnig to that for the two-family case. Furthermore, alike to the case of an initial vanishing sinθ at the renormalizable level for two families, it cannot be corrected by non-renormalizable terms in the potential.

As in section 2.1for two generation, we studied the contributions of non-renormalizable operators in the scalar potential, with similar conclusion: the introduction of higher order terms does not lead to a more natural description of the physical parameters. Nevertheless, some improvement can be obtained when discussing the scenario with a fine-tuned choice of parameters gud, ˜µi. In this case, in fact, lighter Yukawas can be introduced through higher order operators, even if no natural hierarchy between the first two families can be obtained.

In summary, for three generations, to consider bi-fundamental scalars (as in the case ofd = 5 Yukawa operator) alone as the possible dynamical origin of Yukawa couplings does not lead naturally to a satisfactory pattern of masses and mixings.¹⁵

3.2 d = 6 Yukawa operator: the fundamental approach

We deal now with the case of flavons transforming in the fundamental of the flavour group Gf. For most of the conventions we refer to the two-family treatment done in section2.2.

To account for non-trivial mixing, it is necessary to introduce at least four flavons, corre- sponding to up and down, left and right flavons:

χ^Lu ∈ (3, 1, 1) , χ^Ru ∈ (1, 3, 1) , χ^Ld ∈ (3, 1, 1) , χ^Rd ∈ (1, 1, 3) . (3.7) When they develop vevs, the flavour symmetry is spontaneously broken and the Yukawa matrices are given as in eq. (1.9). Without loss of generality, it is possible to write:

hχ^L,R_u,di ≡   χ

L,R u,d

  V_L,R^(u,d)





 0 0 1





 , (3.8)

where V_L,R^(u,d) are 3 × 3 unitary matrices. Similarly to what was shown in section 2.2, removing the unphysical parameters, the following expressions for the Yukawa matrices are obtained:

YD =  χ^L_d

  χ^R_d

 Λ²_f





 0 0 0 0 0 0 0 0 1





 , YU =

 χ^L_u

  χ^R_u

Λ²_f V_L^(d)†V_L^(u)





 0 0 0 0 0 0 0 0 1





 . (3.9) This illustrates that, independently of the parametrization chosen, YD and YU can have only one non-vanishing eigenvalue, as they result from multiplying two vectors. For obvious

15See note added in proof.